📐 geometry

1. **Problem statement:** Two towers of heights 30 m and 40 m are 50 m apart. A well is located between them on the line joining their bases. Two birds fly simultaneously from the

1. **Problem statement:** Given a circle with center O, chord PQ of length 8 cm, N is the midpoint of PQ, ON = 3 cm, and \(\angle ONR = 20^\circ\). We need to find the size of \(\a

1. The problem asks for the total area of a rectangular rug with width 8 meters and height 2 7/12 meters. 2. First, convert the mixed number height to an improper fraction:

1. The problem asks to translate shape A, a trapezoid located approximately between $x = -7$ to $x = -3$ and $y = 3$ to $y = 6$, by the vector $\begin{pmatrix}8 \\ -2\end{pmatrix}$

1. **State the problem:** We are given quadrilateral RSTU with angles and side lengths, and we need to find the measures of angles $\angle RST$ and $\angle RTU$ using the given inf

1. **Stating the problem:** We have a quadrilateral R S T U with diagonal S U. Given angles are \(\angle R = 30^\circ\) and \(\angle R S U = 25^\circ\). The segment \(R U = 4b - 1\

1. **State the problem:** We have quadrilateral RSTU with given angles and side lengths. We need to find the measures of angles $\angle RST$ and $\angle RTU$.

1. The problem asks to identify all triangles similar to triangle T. 2. Similar triangles have the same shape but may differ in size; they have equal corresponding angles and propo

1. **State the problem:** We are given the curved surface area of a hemisphere as 154 and need to find its diameter. 2. **Recall the formula:** The curved surface area (CSA) of a h

1. The problem asks to identify three true statements about the points, lines, and planes based on the given diagram description. 2. From the description:

1. **State the problem:** We need to find the total area of a figure composed of a triangle and a parallelogram. 2. **Identify dimensions:**

1. **State the problem:** We need to find the perimeter and area of a polygon composed of a rectangle and a right triangle attached to its right side. 2. **Identify given dimension

1. The problem states that two right triangles are similar, with Ariadne's height and shadow length given as 5 ft and 15 ft respectively, and Dixon's shadow length as 18 ft. We nee

1. The problem asks to classify a triangle based on its side lengths and angles. 2. The triangle has two equal sides, indicated by the markings, so it is an isosceles triangle.

1. Construct triangle PQR with $PQ=6.5$ cm, $\angle PQR=105^\circ$, and $QR=8.0$ cm. Measure $PR$.\n\nStep 1: Draw segment $PQ=6.5$ cm using a ruler.\nStep 2: At point $Q$, use a p

1. The problem asks for the size of an interior angle of a regular polygon, specifically a regular hexagon which has 6 equal sides. 2. The formula for the measure of each interior

1. **Problem statement:** Given pyramid S.ABCD with trapezoid base ABCD where AD \parallel BC and AD = 3BC. O is intersection of diagonals AC and BD. Points E on SA and F on SD sat

1. The problem asks us to find the area of a sector of a circle with a central angle of 58° and radius 8.1 cm. 2. The formula for the area of a sector is $$\text{Area} = \frac{\the

1. The problem states that a circle has a diameter of 28 m and asks to calculate its area to 3 significant figures. 2. Recall the formula for the area of a circle: $$A = \pi r^2$$

1. The problem asks for the area of a triangle with a base of 4.1 mm and a height of 9.2 mm. 2. The formula for the area of a triangle is $$\text{Area} = \frac{1}{2} \times \text{b

1. **Problem statement:** Given pyramid S.ABCD with trapezoid base ABCD where AD \parallel BC and AD=3BC. O is intersection of diagonals AC and BD. Points E on SA with SE=2EA and F